Saliency-based position estimation in permanent magnet synchronous motors

ABSTRACT

A method and system for the sensorless estimation of the states of the mechanical subsystem of a motor are disclosed. In particular, the method and system split the estimation problem into two sub-problems: the estimation of motor parameters as a function of position, and the determination of the states of the mechanical subsystem of the motor using the estimated parameters via a state observer. Motor current and PWM voltage in a polyphase system is measured or otherwise determined and converted into current and voltage in α-β coordinates. A least square estimator is constructed that uses the voltage and current in α-β coordinates and provides an estimate of the motor parameters, and in particular, the inductances of the motor, which are a function of the rotor position. These estimated inductances are used to drive a state observer whose dynamics have been selected to provide as an output the state of the mechanical subsystem of the motor, and in particular, the position, velocity, and acceleration of this subsystem.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the priority of U.S. Provisional PatentApplication No. 60/366,047 filed, Mar. 20, 2002 entitled SALIENCY-BASEDPOSITION ESTIMATION IN PM SYNCHRONOUS MOTORS, the whole of which ishereby incorporated by reference herein.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Part of the work leading to this invention was carried out with UnitedStates Government support provided under a grant from the Office ofNaval Research, Grant No. N00014-97-1-0704 and under a grant from theNational Science Foundation, Grant No. EC9502636 Therefore, the. U.S.Government has certain rights in this invention.

BACKGROUND OF THE INVENTION

This invention relates generally to motor position estimation systemsand in particular to motor position estimation systems for permanentmagnet synchronous motors.

Permanent magnet synchronous motors (PMSMs) have become increasinglypopular in high-performance variable-frequency drives due to their highefficiency, high torque-to-inertia ratio, rapid dynamic response, andrelatively simple modeling and control. To achieve the proper fieldposition orientation in motion control applications, it is necessary toobtain the actual position of the rotor magnets. Typically, thisposition is obtained through the use of position sensors mounted on themotor shaft. These position sensors can include a shaft encoder,resolver, Hall effect sensor, or other form of angle sensor. Thesesensors typically are expensive, require careful installation andalignment, and are fragile. Accordingly, the the use of these positionsensors both in terms actual cost to produce the product and in thecosts associated with the maintenance and repair of these systems ishigh.

Sensorless position estimation systems have been developed to overcomethese limitations, but the present sensorless position estimationsystems have their own drawbacks as well. These systems typically fallinto two categories: back emf methods and signal injection methods. Backemf systems use the measured back emf voltage to estimate the positionof the rotor magnets. The back emf voltage signal is roughlyproportional to the rotational velocity of the motor. Although thismethod achieves valid results at higher rotational velocities, at lowrotational velocities and high torque settings, the back emf voltagesignal decreases to values that are too small to be useful, and vanishentirely at speeds close to a standstill. Thus, the back emf method isnot suitable for these low rotational velocity high torque applications.The second method is the signal injection method in which an auxiliarysignal is introduced into the motor electrical subsystem and measure theresponse of the motor electrical subsystem to estimate the rotorposition. Although this method avoids the singularities of the back emfmethod, these methods are computationally intensive and thus are notwell suited for industrial applications.

Accordingly, it would be advantageous to provide a position estimationsystem that avoids the singularities at low speeds and thecomputationally intensive methods of the prior art.

BRIEF SUMMARY OF THE INVENTION

A method and system for the sensorless estimation of the states of themechanical subsystem of a motor are disclosed. In particular, the methodand system split the estimation problem into two sub-problems: theestimation of motor parameters as a function of position, and thedetermination of the states of the mechanical subsystem of the motorusing the estimated parameters via a state observer. Motor current andPWM voltage in a polyphase system is measured or otherwise determinedand converted into current and voltage in α-β coordinates. A leastsquare estimator is constructed that uses the voltage and current in α-βcoordinates and provides an estimate of the motor parameters, and inparticular, the inductances of the motor, which are a function of therotor position. These estimated inductances are used to drive a stateobserver whose dynamics have been selected to provide as an output thestate of the mechanical subsystem of the motor, and in particular, theposition, velocity, and acceleration of this subsystem.

Other features, functions, and aspects of the invention will be evidentfrom the Detailed Description of the Invention that follows.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

The invention will be more fully understood with reference to thefollowing Detailed Description of the Invention in conjunction with thedrawings of which:

FIG. 1 is a block diagram of an embodiment of the sensorless positionestimation system;

FIG. 2 is a set of graphs depicting voltage and current waveforms inconjunction with the embodiment of the sensorless position estimatordepicted in FIG. 1;

FIG. 3 is a PWM pattern constellation suitable for use with thesensorless position estimation system depicted in FIG. 1; and

FIG. 4 is a method for performing an embodiment of the sensorlessposition estimation.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 depicts a block diagram of one embodiment of the sensorlessposition estimation system (“system”). The system 100 includes acontroller 102 providing 3 phase PWM control signals to a voltageinverter 104 that is responsive to the PWM control signals by providingthe appropriate phasing to the three currents 101, 103, 105 to the motor106. The three currents 101, 103, and 105 are sensed by current sensor107 and provided to a 3-phase to α-β converter module 108. The convertermodule 108 transforms the 3-phase currents into 2 phases and providesi_(α) and i_(β) values using the well standard transformation:$\begin{bmatrix}y_{\alpha} \\y_{\beta}\end{bmatrix} = {{\sqrt{\frac{2}{3}}\begin{bmatrix}1 & \frac{1}{2} & {- \frac{1}{2}} \\0 & \sqrt{\frac{3}{2}} & {- \sqrt{\frac{3}{2}}}\end{bmatrix}}\quad\begin{bmatrix}x_{1} \\x_{2} \\x_{3}\end{bmatrix}}$where y_(α) and y_(β) are the output in α-β coordinates and x₁, x₂, andx₃ are the 3 polyphase current values. The α-β currents, y_(α)and y_(β),are then provided to the least square parameter estimator 110.

The controller 102 also provides the 3-phase PWM control signals toanother 3-phase to α-β converter module 112 that converts the 3-phasevoltage commands into α-β coordinates using the above transformation inwhich x₁, x₂, and x₃ are the 3 polyphase voltage signals and y_(α) andy_(β) are the voltages v_(α) and v_(β) and are provided to the parameterestimator 110. The parameter estimator 110 uses the transformed currentand voltage signals to estimate a set of parameters for the motor 106.These estimated parameters are provided as an input to the mechanicalobserver 114 that uses the estimated parameters to provide estimates ofthe position, velocity, and acceleration of the mechanical subsystem ofmotor 106.

The magnetic field of a motor with salient rotors changes along thecircumference. In addition to this non-isotropic behavior due to rotorgeometry, in permanent magnetic synchronous motor (PMSMs) additionalasymmetry is introduced by the non-uniform magnetic saturation of thestator steel. Due to these phenomena, the fluxes through the statorwindings produced by the phase currents are a function of the rotorposition that implies a position dependence of the stator phaseinductances. The resulting stator inductance matrix in a stationary α-βframe is given by: $\begin{matrix}{L = \begin{bmatrix}{L_{0} + {L_{1}{\cos\left( {2\theta} \right)}}} & {L_{1}{\sin\left( {2\theta} \right)}} \\{L_{1}{\sin\left( {2\theta} \right)}} & {L_{0} - {L_{1}{\cos\left( {2\theta} \right)}}}\end{bmatrix}} & (1)\end{matrix}$The equations describing the electrical subsystem of the motor in theα-β frame are: $\begin{matrix}\begin{matrix}{v = {{R_{s}i} + \frac{\mathbb{d}\lambda}{\mathbb{d}t}}} \\{\lambda = {{Li} + \lambda_{\tau}}}\end{matrix} & (2)\end{matrix}$where v=[v_(α)v_(β]) ^(T) and i=[i_(α)i_(β)]^(T) are the vectors of thestator phase voltages and currents, λ=[λ_(α)λ_(β]) ^(T) andλ_(T)=[λ_(τα)λ_(τβ)]^(T) are the vector of stator phase total andpartial fluxes, which are due to the rotor field and R_(s)=diag[Rs, Rs]is the matrix of the stator phase resistances. Combining the aboveequations yields the following electrical subsystem in the α-β frame:$\begin{matrix}{v = {{\left( {R_{s} + {\omega\frac{\mathbb{d}L}{\mathbb{d}\theta}}} \right)i} + {L\frac{\mathbb{d}i}{\mathbb{d}t}} + {\omega\frac{\mathbb{d}\lambda_{r}}{\mathbb{d}\theta}}}} & (3)\end{matrix}$where $\begin{matrix}{R_{s} = \begin{bmatrix}{R_{s} - {2\omega\quad L_{1}{\sin\left( {2\theta} \right)}}} & {2\omega\quad L_{1}{\cos\left( {2\theta} \right)}} \\{2\omega\quad L_{1}{\cos\left( {2\theta} \right)}} & {R_{s} + {2\omega\quad L_{1}{\sin\left( {2\theta} \right)}}}\end{bmatrix}} & (4)\end{matrix}$is the effective resistance matrix.

The torque produced by the motor includes reluctance and mutualcomponents and, neglecting torque ripple, is given by: $\begin{matrix}{\tau_{m} = {P\left( {{\frac{1}{2}i^{T}\frac{\mathbb{d}L}{\mathbb{d}\theta}i} + {i^{T}\frac{\mathbb{d}\lambda_{r}}{\mathbb{d}\theta}}} \right)}} & (5)\end{matrix}$where P is the number of pole pairs. The mechanical behavior of themotor is modeled with the viscous friction and inertia as:$\begin{matrix}\begin{matrix}{{J\frac{\mathbb{d}\omega}{\mathbb{d}t}} = {\tau_{m} - {B\quad\omega} - \tau_{l}}} \\{\frac{\mathbb{d}\theta}{\mathbb{d}t} = \omega}\end{matrix} & (6)\end{matrix}$where J and B are the moment of inertia and the friction constantnormalized with respect to P, and τ₁ is the load torque.

The position estimation system utilizes the above model and uses themechanical state variables as parameters in the inductance andresistance matrices. This method uses the electrical transients, whichhave a much faster time constant than the mechanical transients, toestimate the approximately constant parameters of the system. Theelectrical subsystem is given by: $\begin{matrix}{v = {{Ri} + {L\frac{\mathbb{d}i}{\mathbb{d}t}} + {\omega\frac{\mathbb{d}\lambda_{\tau}}{\mathbb{d}\theta}}}} & (7)\end{matrix}$where ${L = \begin{bmatrix}{L_{0} + {L_{1}{\cos\left( {2\theta} \right)}}} & {L_{1}{\sin\left( {2\theta} \right)}} \\{L_{1}{\sin\left( {2\theta} \right)}} & {L_{0} - {L_{1}{\cos\left( {2\theta} \right)}}}\end{bmatrix}},\quad{R = \begin{bmatrix}{R_{s} - {2\omega\quad L_{1}{\sin\left( {2\theta} \right)}}} & {2\omega\quad L_{1}{\cos\left( {2\theta} \right)}} \\{2\omega\quad L_{1}{\cos\left( {2\theta} \right)}} & {R_{s} + {2\omega\quad L_{1}{\sin\left( {2\theta} \right)}}}\end{bmatrix}}$are the inductance and effective resistance matrices, where${L_{0} = \frac{L_{d} + L_{q}}{s}},{L_{1} = {\frac{L_{d} - L_{q}}{2}.}}$

FIG. 2 depicts representative current and voltage waveforms in a singlePWM period consisting of 1, 2, 3, . . . N subintervals. It is assumedthat the voltage v_(α) in graph 2(b) is constant over the entiresubinterval and that the current in graph 2(a) is substantially linearover the entire subinterval. Thus, the integration of equation (7) overthe (t_(n+1), t_(n)) interval and division by this interval todiscretize equation (7) yields: $\begin{matrix}{v_{n} = {{R\frac{i_{n + 1} + i_{n}}{2}} + {L\frac{i_{n + 1} + i_{n}}{t_{n + 1} - t_{n}}} + {\omega\frac{\mathbb{d}\lambda_{\tau}}{\mathbb{d}\theta}}}} & (8)\end{matrix}$where n=1, 2, . . . N, and N is a number of subintervals within one PWMperiod. Assuming that the mechanical parameters do not change from onesubinterval to another, the back emf term, i.e., the ω term in equation(8) can be removed by subtracting two subsequent subintervals whichyields: $\begin{matrix}{{v_{n + 2} - v_{n + 1}} = {{L\left( {\frac{i_{n + 2} + i_{n + 1}}{t_{n + 2} - t_{n + 1}} - \frac{i_{n + 1} + i_{n}}{t_{n + 1} - t_{n}}} \right)} + {R\left( \frac{i_{n + 2} + i_{n + 1}}{2} \right)}}} & (9)\end{matrix}$At the PWM frequencies that are contemplated for use in this system, thecontribution of the R term in equation (9) is negligible, andaccordingly the term is neglected. This reduces the problem to be solvedto: $\begin{matrix}{w_{n} = {{Lx}_{n}\quad{where}}} & (10) \\\begin{matrix}{w_{n} = {v_{n + 1} - v_{n}}} \\{x_{n} = \left( {\frac{i_{n + 2} + i_{n + 1}}{t_{n + 2} - t_{n + 1}} - \frac{i_{n + 1} + i_{n}}{t_{n + 1} - t_{n}}} \right)}\end{matrix} & (11)\end{matrix}$where x_(n) is the slope of the current is determined.

To solve the parameter estimation problem, a least squares problem isconstructed, but to simplify the process of solving the least squaresproblem, equation (10) is rewritten such that unknown matrix containingthe parameters to be estimated multiplied by the already known voltagevalues as: $\begin{matrix}{x_{n} = {L^{- 1}w_{n}\quad{where}}} & (12) \\{L^{- 1} = {\frac{1}{L_{0}^{2} - L_{1}^{2}}\begin{bmatrix}{L_{0} - {L_{1}{\cos\left( {2\theta} \right)}}} & {{- L_{1}}{\sin\left( {2\theta} \right)}} \\{{- L_{1}}{\sin\left( {2\theta} \right)}} & {L_{0} + {L_{1}{\cos\left( {2\theta} \right)}}}\end{bmatrix}}} & (13)\end{matrix}$A parameterization to be used in the estimation problem is given by:$\begin{matrix}{q = {\begin{bmatrix}q_{0} \\q_{1} \\q_{2}\end{bmatrix} = {\frac{1}{L_{0}^{2} - L_{1}^{2}}\begin{bmatrix}L_{0} \\{{- L_{1}}{\cos\left( {2\theta} \right)}} \\{{- {L1}}\quad{\sin\left( {2\theta} \right)}}\end{bmatrix}}}} & (14)\end{matrix}$Accordingly, equation (11) can be rewritten as: $\begin{matrix}{x_{n} = {\begin{bmatrix}w_{n\quad\alpha} & w_{n\quad\alpha} & w_{n\quad\beta} \\w_{n\quad\beta} & {- w_{n\quad\beta}} & w_{n\quad\alpha}\end{bmatrix}\quad\begin{bmatrix}q_{0} \\q_{1} \\q_{2}\end{bmatrix}}} & (15)\end{matrix}$for n=1,2,3, . . . N−1.

Thus, equation (14) can be writtenx=Wq  (16)where W is the matrix of the n−1 voltage differences between adjacentPWM subintervals for the α-β coordinates from equation (11).

Stacking the N−1 equations to form the least squares problem ensuresthat the system in equation (15) is over-determined, the solution forthe estimate of q is given by:{circumflex over (q)}=(W ^(T) W)⁻¹ W _(T) x=W _(pi) x  (17)

In order to further simplify this problem, a particular PWM pattern isused. As is known, the PWM signal is a three-bit binary signal providedby the controller 102 to the voltage inverter 104 and where a “1”indicates that the corresponding phase is connected to the voltage railand a “0” indicates that the corresponding phase is connected to ground.As discussed above, the basic unit of time is referred to as a PWMperiod that is subdivided into N subintervals during which a particularinput switch combination is held constant. The duration of eachsubinterval, referred to as the “duty cycle” can vary, as can the switchcombination of each subinterval. In general, the vectors and duty cyclesare selected to produce the desired PWM output vector. In the embodimentdescribed herein the PWM pattern has six subintervals and the switchingcombination is always each of the six non-zero PWM vectors (V₁ . . .V₆), see FIG. 3 for a graphical depiction of the PWM vectors. The outputPWM vector is therefore dependent upon the duty cycle of eachsubinterval which are computed according to: $\begin{matrix}{{{\varsigma_{1} = {\frac{1}{6} + {\frac{5\sqrt{3}}{12}\frac{v_{a}}{v_{\max}}} - {\frac{7}{12}\frac{v_{\beta}}{v_{\max}}}}}\varsigma_{2} = {\frac{1}{6} - {\frac{\sqrt{3}}{12}\frac{v_{a}}{v_{\max}}} + {\frac{11}{12}\frac{v_{\beta}}{v_{\max}}}}}{\varsigma_{3,4,5,6} = {\frac{1}{6} - {\frac{\sqrt{3}}{12}\frac{v_{a}}{v_{\max}}} - {\frac{1}{12}\frac{v_{\beta}}{v_{\max}}}}}} & (18)\end{matrix}$where the duty ratios are defined as${\varsigma_{i} = \frac{t_{i}}{T_{PWM}}},\varsigma_{1}$is the duty ratio of the subinterval corresponding to the lag vector,i.e., one of the vectors adjacent to the desired vector, ζ₂ is the dutyratio of the subinterval corresponding to the lead vector, i.e., theother one of the vectors adjacent to the desired vector, and ζ_(3,4,5,6)correspond to the other vectors.

This choice of PWM pattern that is consistent in the order in which thePWM vectors are processed and in which the value of each is consistentallows W in equation (17) to be constant over time since the voltagelevels in each subinterval are the same. Thus, W_(pi) can be computedoff-line and {circumflex over (q)}, the estimate of the parameters, willbe reduced to a multiplication of a constant matrix by a vector thatdepends on the measured current values and the duration of thesubintervals.

Once {circumflex over (q)} has been estimated, in one embodiment therotor position could be computed using inverse trigonometric functionsand solving equation (13) for θ. However, this has at least twodisadvantages. First, the inverse trigonometric functions can becomputationally intensive to calculate, and second, the parameterestimate is noisy and would not be filtered. Accordingly, to avoid theseproblems, in the preferred embodiment, a state estimator is providedthat will provide an estimate for all the mechanical states of thesystem. The state estimator provides for filtering of the parameters andis not computationally intensive. The dynamics of the state observer aredesigned to be much faster than the dynamics of the mechanical loopcontroller, so that it is neglected during control loop design.

The observer has the following dynamic form:{circumflex over({dot over (α)})}=γ₃ε{circumflex over({dot over (ω)})}=γ₂ε+{circumflex over (α)}{circumflex over({dot over (θ)})}=γ₁ε+{circumflex over (ω)}  (19)where {circumflex over (θ)},{circumflex over (ω)}, and{circumflex over(α)} are the estimates of the position, speed, and accelerationrespectively, γ_(I) are design parameters and ε is a non-linearobservation error that drives the variable estimates to their truevalue. The non-linear observation error is derived from the inductanceparameters as:={circumflex over (p)} ₃ cos(2{tilde over (θ)})−{circumflex over (p)} ₂sin(2{tilde over (θ)})=L ₁ sin(2θ)  (20)where {tilde over (θ)}=θ−{circumflex over (θ)}. It can be shown that theerror dynamics of the state observer are non-linear with stableequilibria at {tilde over (θ)}=nπ and if linearized around 0 the errordynamics are stable and the values provided by the state observer willbe driven toward the correct values. The speed of the response of thestate observer is determined by the placement of the poles of the systemwhich are set by the values selected for γ_(i).

The selection of the poles presents a tradeoff between the response timeof the state observer and noise filtering since the bandwidth increasesfor a faster response time but with less filtering and vice-versa.Another consideration of the pole placement is to keep the originalstate observer system in equation (19) in the neighborhood of {tildeover (θ)}=0 under all operating conditions.

Accordingly, for each update of the estimated parameters, the estimatedparameters are provided to the state observer in equation (19). Equation(19) is then solved and provides an estimate vector of the position,speed, and acceleration of the mechanical subsystem of the motor for theprevious PWM subinterval.

FIG. 4 depicts a method for performing an embodiment of the sensorlessposition estimation system. In step 402, the W matrix is formed based onthe PWM pattern as discussed above. In step 404, the current in eachphase is measured. In step 406, the three phases of current areconverted into the α-β frame as discussed above. In step 408, the vectorx is formed and the slope of the current is determined as discussedabove. In step 410, the estimated parameters are determined according toequation (16). In step 412, the newly estimated parameters are providedto the state observer, equation (18), and the state observer is updated.In step 414 the estimated mechanical parameters, i.e., the position,velocity, and acceleration of the mechanical subsystem are output.

It should be appreciated that other variations to and modifications ofthe above-described sensorless position estimation system may be madewithout departing from the inventive concepts described herein.Accordingly, the invention should not be viewed as limited except by thescope and spirit of the appended claims.

1. An apparatus for providing a position estimate of a polyphase motorreceiving polyphase current from a voltage inverter, the voltageinverter receiving PWM commands from a controller, the apparatuscomprising: a current measuring apparatus for measuring at least twopolyphase currents; an α-β converter coupled to the current measuringapparatus, the α-β converter operative to convert the polyphase currentsinto the α-β frame of reference, the α-β converter providing an outputof the current in α-β coordinates; the α-β converter further coupled tothe controller and receiving the PWM commands therefrom, the α-βconverter operative to convert the PWM commands into α-β coordinates andto provide as an output the PWM commands in α-β coordinates; a leastsquare estimator, receiving the current in α-β coordinates and the PWMcommands in α-β coordinates, the least square estimator operative toprovide an output vector {circumflex over (q)} of estimated parameters,the estimated parameters including estimated inductances of the motor;and a state observer coupled to the least square estimator and receivingthe estimated parameters including the estimated inductances of themotor therefrom, the state observer operative to provide an output of anestimate of a mechanical subsystem of the motor that includes a positionestimate of the mechanical subsystem of the motor.
 2. The apparatus ofclaim 1 wherein the least square estimator estimates the output vector{circumflex over (q)} of estimated parameters by solving an equationq=(W ^(T) W)⁻¹ W ^(T) x, wherein W is a matrix of n−1 voltagedifferences between adjacent PWM subintervals of a PWM period, and x isa vector of the polyphase currents.
 3. The apparatus of claim 1 whereinthe state observer is of the form:{circumflex over({dot over (α)})}=γ₃ε,{circumflex over({dot over (ω)})}=γ₂ε+{circumflex over (α)}, and{circumflex over({dot over (θ)})}=γ₁ε+{circumflex over (ω)}, wherein{circumflex over (θ)} is the position estimate, {circumflex over (ω)} isa speed estimate, and {circumflex over (α)} is an acceleration estimateof the mechanical subsystem of the motor, γ^(I) are design parametersfor setting poles of the subsystem to determine a speed of response ofthe state observer, and ε is a non-linear observation error of the formε=L ₁ sin (2θ), wherein θ is a position of the mechanical subsystem ofthe motor, and $L_{1} = \frac{L_{d} - L_{q}}{2}$ wherein L₁ representsthe estimated inductances of the motor, L_(d) denoting inductance in amagnetic flux direction of the polyphase currents, and L_(q) denotinginductance in a direction perpendicular to the magnetic flux direction.4. A method for providing a position estimate of a polyphase motorreceiving polyphase current from a voltage inverter, the voltageinverter receiving PWM commands from a controller, the method comprisingthe steps of: forming a matrix W, wherein the matrix W is a matrix ofn−1 voltage differences between adjacent PWM subintervals of a PWMperiod; obtaining measurements of at least two polyphase currents;converting the obtained current measurements into an α-β coordinatesystem; forming a vector x, wherein the vector x is a vector of thepolyphase currents; computing an output vector {circumflex over (q)} ofestimated parameters using an equation {circumflex over(q)}=(W^(T)W)⁻¹W^(T)x; providing the computed output vector {circumflexover (q)} of estimated parameters to a state observer; updating thestate observer by solving{circumflex over({dot over (α)})}=γ₃ε,{circumflex over({dot over (ω)})}=γ₂ε+{circumflex over (α)}, and{circumflex over({dot over (θ)})}=γ₁ε+{circumflex over (ω)}, wherein{circumflex over (θ)} is the position estimate, {circumflex over (ω)} isa speed estimate, and {circumflex over (α)} is an acceleration estimateof a mechanical subsystem of the motor, γ_(I) are design parameters forsetting poles of the subsystem to determine a speed of response of thestate observer, and ε is a non-linear observation error of the formε=L ₁ sin(2θ), wherein θ is a position of the mechanical subsystem ofthe motor, and ${L_{1} = \frac{L_{d} - L_{q}}{2}},$ and wherein L_(d)denotes inductance in a magnetic flux direction of the polyphasecurrents, and L_(q) denotes inductance in a direction perpendicular tothe magnetic flux direction; and providing an output of an estimate ofthe mechanical subsystem of the motor that includes a position estimateof the mechanical subsystem of the motor.